Symplectic Geometry
In this chapter, we study symplectic manifolds. We start with the Theorem of Darboux, which states that all symplectic structures of a given dimension are locally equivalent. Thus, in sharp contrast to the situation in Riemannian geometry, symplectic mani
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Symplectic Geometry
Symplectic geometry plays a tremendous role both in pure mathematics and in physics. As we will see, it provides the natural mathematical language for the study of Hamiltonian systems. In this chapter, we present the basic notions of symplectic geometry. The starting point will be the Theorem of Darboux, which states that, locally, all symplectic structures of a given dimension are equivalent to the standard symplectic vector space structure on R2n defined in the previous chapter. Thus, in sharp contrast to the situation in Riemannian geometry, there are no local invariants. Symplectic manifolds of the same dimension can at most differ globally.1 According to the above equivalence, many structures of local symplectic geometry have their origin in symplectic algebra. The second elementary, but very important observation is that on symplectic manifolds the symplectic form provides a duality between smooth functions and certain vector fields, called Hamiltonian vector fields. As an immediate consequence of this duality, we obtain the notion of Poisson structure. Given the great importance of Poisson structures both in mathematics and in physics, we go beyond the symplectic case and give a brief introduction to general Poisson manifolds. There are two classes of symplectic manifolds which are especially important in physical applications: cotangent bundles and orbits of the coadjoint representation of a Lie group. We will see that the cotangent bundle serves as a mathematical model of phase space and coadjoint orbits are relevant in the study of systems with symmetries. Both classes of examples are discussed in detail. Moreover, in this chapter we discuss elementary properties of coisotropic submanifolds, present a number of natural generalizations of the Darboux Theorem and give an introduction to general symplectic reduction. The important special case of symplectic reduction for systems with symmetries will be dealt with in Chap. 10. The more advanced theory of Lagrangian submanifolds (including topological as1 There is a huge field of research called symplectic topology, which deals with the study of global invariants of symplectic manifolds. For a nice intuitive introduction to this field we refer the reader to an article of Arnold [22]. There is a number of detailed expositions of this subject, see e.g. [206].
G. Rudolph, M. Schmidt, Differential Geometry and Mathematical Physics, Theoretical and Mathematical Physics, DOI 10.1007/978-94-007-5345-7_8, © Springer Science+Business Media Dordrecht 2013
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Symplectic Geometry
pects and the study of singularities) is contained in Chap. 12, where it finds one of its natural applications. Finally, the last section of this chapter is devoted to an elementary introduction to Morse theory. As we will see, the basic notions of this theory can be naturally formulated in the language of symplectic geometry. Moreover, concepts of Morse theory are of special importance in the study of symplectic manifolds and Hamiltonian systems. This is relate
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